# Broadcast Product: Redefining Shape-aligned Element-wise Multiplication and Beyond

arXiv: https://arxiv.org/abs/2409.17502 · HTML: https://mlsanity.com/paper/2409.17502/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 26 Sep 2024.

## Builds on (most cited first, 17 of 17)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Arbitrary Style Transfer in Real-time with Adaptive Instance Normalization](https://mlsanity.com/paper/1703.06868/) (2017)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Instance Normalization: The Missing Ingredient for Fast Stylization](https://mlsanity.com/paper/1607.08022/) (2016)
- [DynIBaR: Neural Dynamic Image-Based Rendering](https://mlsanity.com/paper/2211.11082/) (2022)
- [Real-Time High-Resolution Background Matting](https://mlsanity.com/paper/2012.07810/) (2020)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [Formal Algorithms for Transformers](https://mlsanity.com/paper/2207.09238/) (2022)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives](https://mlsanity.com/paper/1708.09165/) (2017)
- [Alternating Local Enumeration (TnALE): Solving Tensor Network Structure Search with Fewer Evaluations](https://mlsanity.com/paper/2304.12875/) (2023)
- [Named Tensor Notation](https://mlsanity.com/paper/2102.13196/) (2021)
- [An introduction to graphical tensor notation for mechanistic interpretability](https://mlsanity.com/paper/2402.01790/) (2024)
- [Efficient ConvBN Blocks for Transfer Learning and Beyond](https://mlsanity.com/paper/2305.11624/) (2023)

## Built on by (most cited first, 1 of 1)

- [Very Basics of Tensors with Graphical Notations: Unfolding, Calculations, and Decompositions](https://mlsanity.com/paper/2411.16094/) (2024)

## Abstract (arXiv)

Broadcast operations are widely used in scientific computing libraries, yet their mathematical formulation is often implicit and inconsistently represented in machine learning literature. This problem frequently leads to invalid equations when element-wise products are written despite mismatched tensor shapes. In this paper, we formalize such operations by introducing the broadcast product boxdot, which explicitly extends the Hadamard product through shape-aligned element duplication. We provide a rigorous definition of the broadcast product, analyze its algebraic properties, and show how it can be expressed using standard linear algebra. Building on this framework, we formulate least-squares problems and sketch a proof-of-concept broadcast decomposition. As a preliminary illustration, we show that the formalism enables a new family of decompositions with distinct structural properties from conventional tensor decompositions. This work establishes a mathematical foundation for broadcast-aware tensor operations, connecting practical implementations with rigorous tensor analysis.